Centralizers of Subsystems of Fusion Systems

Abstract

When $(S,\mathcal{F},\mathcal{L})$ is a $p$-local finite group and $(T,\mathcal{E},\mathcal{\L}_0)$ is weakly normal in $(S,\mathcal{F},\mathcal{L})$ we show that a definition of $C_S(\mathcal{E})$ given by Aschbacher has a simple interpretation from which one can deduce existence and strong closure very easily. We also appeal to a result of Gross to give a new proof that there is a unique fusion system $C_{\mathcal{F}}(\mathcal{E})$ on $C_S(\mathcal{E})$.